Solve the System
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Most people are familiar with the traditional methods of solving single equations in one unknown, be they linear, quadratic, or of higher order, as well as sets of equations with multiple unknowns. The solution of simultaneous equations is a relatively simple process given that there are as many equations as there are unknowns. There are situations however, especially in the broad field of recreational mathematics, when the number of equations is less than the number of unknowns. For instance, if there is one less equation than there are unknowns, the best one can do is reduce the set of equations to one equation with two unknowns. This type of equation often forms the basis of solving a wide variety of recreational mathematics problems. This type of equation typically takes the form ax + by = c and is typically referred to as a linear indeterminate algebra problem or a linear Diophantine problem.
There are methods of solving such equations that are relatively straight forward. Aside from the trial and error approach, two methods most commonly used are the Euclidian Algorithm and the method of repeated reductions.
The Euclidian Algorithm is fundamentally an effective method for determining the greatest common divisor/factor of two numbers, a and b, expressed as g.c.d.(a,b). It is also an effective tool in solving linear indeterminate algebra problems. The detailed derivation and proof of the method is much too long to present here and I refer you to any good book on Number Theory to obtain an understanding of its background and derivation.
Given:
1--X1 + X2 + 4X3 = -7
2--4X1 + 5X2 – 3X3 = 1
3--Multiplying (1) by 3, (2) by 4 and adding yields 19X1 + 23X2 = -17
4--Solve 19X1 + 23X2 = -1 first and then multiply the results by 17.
5--23 = 1(19) + 4 or 4 = 23 – 1(19)
6--19 = 4(4) + 3 or 3 = 19 – 4(4)
7--4 = 1(3) + 1 indicating that the g.c.d. of 19 and 23 is 1.
8--Substituting (6) into (7) yields 5(4) – 1(19) = 1
9--Substituting (5) into (8) yields 5(23) – 6(19) = 1 or 19(6) + 23(-5) = -1
10--Multiplying by 17 yields 19(102) + 23(-85) = -17
11--Therefore, X1 = 102 and X2 = -85, from which X3 = -6
12--This is clearly not a singularly unique solution.
13--The general solution of X1 and X2 is X1 = 102 – 23t, X2 = -85 + 19t and X3 naturally follows as X3 = t - 6.
.t…...0……1…...2…...3….4…...5…..6…>
X1…102…79…..56.…33…10…-13…-36…>
X2…-85…-66…-47…-28…-9…..10.…29…>
X3…-6…..-5….-4…..-3…-2…..-1…..0…>
Solutions also exist in the negative “t” direction.
If this is of some interest to you, I urge you to do a search for Euclidian Algorithm.
There are methods of solving such equations that are relatively straight forward. Aside from the trial and error approach, two methods most commonly used are the Euclidian Algorithm and the method of repeated reductions.
The Euclidian Algorithm is fundamentally an effective method for determining the greatest common divisor/factor of two numbers, a and b, expressed as g.c.d.(a,b). It is also an effective tool in solving linear indeterminate algebra problems. The detailed derivation and proof of the method is much too long to present here and I refer you to any good book on Number Theory to obtain an understanding of its background and derivation.
Given:
1--X1 + X2 + 4X3 = -7
2--4X1 + 5X2 – 3X3 = 1
3--Multiplying (1) by 3, (2) by 4 and adding yields 19X1 + 23X2 = -17
4--Solve 19X1 + 23X2 = -1 first and then multiply the results by 17.
5--23 = 1(19) + 4 or 4 = 23 – 1(19)
6--19 = 4(4) + 3 or 3 = 19 – 4(4)
7--4 = 1(3) + 1 indicating that the g.c.d. of 19 and 23 is 1.
8--Substituting (6) into (7) yields 5(4) – 1(19) = 1
9--Substituting (5) into (8) yields 5(23) – 6(19) = 1 or 19(6) + 23(-5) = -1
10--Multiplying by 17 yields 19(102) + 23(-85) = -17
11--Therefore, X1 = 102 and X2 = -85, from which X3 = -6
12--This is clearly not a singularly unique solution.
13--The general solution of X1 and X2 is X1 = 102 – 23t, X2 = -85 + 19t and X3 naturally follows as X3 = t - 6.
.t…...0……1…...2…...3….4…...5…..6…>
X1…102…79…..56.…33…10…-13…-36…>
X2…-85…-66…-47…-28…-9…..10.…29…>
X3…-6…..-5….-4…..-3…-2…..-1…..0…>
Solutions also exist in the negative “t” direction.
If this is of some interest to you, I urge you to do a search for Euclidian Algorithm.